On Approximability of l²₂Min-Sum Clustering

The l22 min-sum k-clustering problem P is to partition an input set into clusters C1, . . . ,Ck to minimize (Formula presented). Although l22 min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate l22 min-sum k-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the l22 min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a fast PTAS for l22 min-sum k-clustering. Specifically, our algorithm runs in time O(n1+o(1)d · 2(k/ϵ)O(1)), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error α ∈ [0, 1/2). We give a polynomial-time algorithm that outputs a 1+γα/(1-α)2 -approximation to l22min-sum k-clustering, for a fixed constant γ > 0.